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7. Lagrangian cobordisms from plabic graphs


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      Problem 7.1.

      For each positroid stratum of a Grassmanian, there is a Legendrian link whose augmentation variety is that stratum. Positroid strata are also ordered with respect to closure.

      Is this partial order recoverable on the Legendrian links in terms of topology, for instance by (decomposable) Lagrangian cobordisms? Could the difference in dimension be related to the minimal genus of the cobordisms?
        1. Remark. [org.aimpl.user:awu@lsu.edu] Indeed, they can be related by a decomposable Lagrangian cobordism: https://arxiv.org/abs/2305.16232.


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              Problem 7.2.

              Is there a local operation of plabic graphs which induces Lagrangian cobordisms between the associated Legendrian knots? In general, what can be done on the combinatorial side to guarantee a Lagrangian cobordism on the other side?


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                  Problem 7.3.

                  Is there a symplectic geometric application of 3d-plabic graphs? Do 3d-plabic graphs give some filling of a Legendrian link?


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                      Problem 7.4.

                      For ADE types, all seeds in the cluster algebra can be realized by Lagrangian fillings. Can this be generalized?


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                          Problem 7.5.

                          Can mutations of quivers with potentials be realized as a symplectic operation? Could we use this to write down potentials for quivers from braid varieties?

                              Cite this as: AimPL: Cluster algebras and braid varieties, available at http://aimpl.org/clusterbraid.